The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Cornelia Mihaila.
I will give a brief introduction on the Landau equation and some of the relevant mathematical challenges and results in its analysis. In a joint work with Chris Henderson and Stanley Snelson, we prove local well-posedness with no a priori assumptions on the solution, even if the initial data contains vacuum regions. We also show, using probabilistic techniques, that nontrivial initial data spreads a positive amount of mass to all regions of space and all positive times. I will present some key ideas of the proof of these results, including a deterministic proof of the mass-spreading which is different from the proof in the paper.
Topological insulators are materials whose macroscopic properties ("phase") are characterized by a non-trivial topological invariant. A practical, remarkable, feature of such insulators occurs when two materials with different invariants share an interface. Transport along that interface is then asymmetric, with more "modes" propagating in one direction, say up, than the other, say down. Moreover, the asymmetric transport is itself protected by an interface topological invariant, which makes it immune to all sorts of random perturbations in the composition of the material. This behavior is in striking contrast to the topologically trivial case, where (enough of) randomness leads to (Anderson) wave localization: nothing really propagates up or down in such a situation. A non-trivial topology may therefore be seen as an obstruction to localization.
Such electronic and photonic structures have been built experimentally and are actively researched in many engineering areas for their unusual immunity to the influence of defects, sometimes with some confusion about what is or is not protected topologically. For instance, is back-scattering, a practical nuisance, really topologically protected?
This talk looks at a description of such topological materials that allows us to characterize the influence of a (large) class of random perturbations. It takes the form of a system of massive Dirac equations in two space dimensions, whose topological invariants are described by the indices of Fredholm operators assigned to the Dirac Hamiltonians. We develop a scattering theory describing currents up and down and assess the quantitative influence of random fluctuations on transport along the interface. In an appropriate regime of wave propagation (diffusion regime), we quantify the topological obstruction to localization: a number of (random) modes exactly prescribed by the topological invariant does transport (up) while all other modes localize. This provides a nuanced answer to the above question. Let us mention that the result generalizes to the setting of "fermionic time reversal symmetry'', where the above (Noether/Fredholm) index vanishes and another symmetry-protected ℤ2 index describes topology.
Clouds and precipitation are among the most challenging aspects of weather and climate prediction. New insights have recently appeared from a promising analysis of observational data from a statistical physics perspective. In this talk, we present stochastic models that help to solidify this connection. In particular, it will be shown that stochastic models of water vapor dynamics can reproduce a wide array of observational statistics that characterize clouds and precipitation in terms of critical phenomena and phase transitions. Several interesting mathematical questions and applications to climate will be discussed. As one application, it will be shown that, under global warming, the most significant change in precipitation statistics is seen in the largest events -- which become even larger and more probable -- and the distribution of event sizes conforms to the stochastic models.
This talk is about the Muskat problem with and without surface tension, modeling the filtration of two incompressible immiscible fluids in porous media. We consider the case in which the fluids have different constant densities together with different constant viscosities. In this situation the equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove global in time existence and uniqueness results for medium size initial stable data in critical spaces. In particular we prove for the first time the global in time stability of star shaped bubbles influenced by Gravity. This is joint work with Gancedo, Garcia-Juarez, and Patel.
We prove new comparison principles for viscosity solutions of non-linear integro-differential equations. The operators to which the method applies include but are not limited to those of Levy-Ito type. The main idea is to use an optimal transport map to couple two different Levy measures, and use the resulting coupling in a doubling of variables argument. This is a joint work with Nestor Guillen and Andrzej Swiech.
I will talk about extending my earlier work on the vanishing noise limit of diffusions in noisy heteroclinic networks to longer time scales. In this field, the results are based on sequential analysis of exit locations and exit times for neighborhoods of unstable equilibria. The new results on exit times and the emergent hierarchical structures based on polynomial transition rates rather than exponential ones of the Freidlin-Wentzell metastability theory are joint with Zsolt Pajor-Gyulai.
Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.